Cutting with Boolean Modifiers
You could add a solidify modifier to the sphere before the boolean modifier:



Also note that you can set the boolean modifier to Difference and use a plane as the Object, it may be easier to animate than a cube.

For this to work, the normal of the plane must be facing the sphere:



You can flip the direction of the normals by


Pressing Flip Direction in 3D view > Tool Shelf (N) > Basic > Normals
Pressing CtrlN and enabling Inside in the Redo menu (either at the bottom of the Tool Shelf, or by pressing F6).


Result: